1. Ned confessed in order to save Sansa. He wasn't given a trial like Tyrion at the Eyrie, the deal was if he confessed he would be allowed to take the black and no harm would come of Sansa.

2. It would go against Neds character to choose someone to fight and possibly die for him imo.

3. Selmy wouldn't have fought anyway he was on the run although he was in the crowd at the execution he was dressed as an old beggar, Ned would have no way of knowing he was there and Cersei is unlikely to allow someone on the run to champion Ned. Also there was no one else in Kings Landing that either would or could fight for him.

4. It would take Greatjon or anyone from the North weeks to travel to Kings Landing